Probability calculations for collision and impingement analysis need to ensure sufficient accuracy to give meaningful results. Because all operational decisions are ultimately made with respect to the amount of acceptable risk, the action threshold should not be based on an unacceptable miss distance or keep-out angle but rather on an unacceptable probability. NASA currently uses a risk-based approach with the International Space Station and Space Transportation System, where avoidance maneuvers are initiated if the collision probability becomes unacceptably high. If the positional uncertainty is very large, a Gaussian calculation will produce a low conjunction probability. Although mathematically correct, the resulting probability may give a false sense of confidence that a conjunction is not likely to occur. Such a low probability may, in fact, indicate that the data is not of sufficient accuracy to produce an operationally meaningful result.
For satellite tracking, the accuracy of positional covariance matrices resulting from Least Squares Orbit Determination of sparse data is questionable. Covariance matrices formed in this manner often provide overly optimistic results. Frisbee and Foster noted in “A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles,” NASA JSC 25898, August 1992, that “the primary problem with state error covariances determined from observations of objects in Earth orbit is that they are not truly reflective of the uncertainties in the dynamic environment.” To address this concern, they devised a method to scale covariances provided to NASA by Air Force Space Command. The present invention provides a way to address the inaccuracy of covariance matrices by determining a mathematical upper bound that will not be exceeded regardless of positional uncertainty.
Significant work has been done to address the computing of collision and impingement probability for neighboring space objects (see FOSTER, J. L., and ESTES, H. S., “A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles,” NASA JSC 25898, August 1992; KHUTOROVSKY, Z. N., BOIKOV, V., and KAMENSKY, S. Y., “Direct Method for the Analysis of Collision Probability of Artificial Space Objects in LEO: Techniques, Results, and Applications,” Proceedings of the First European Conference on Space Debris, ESA SD-01, 1993, pp. 491-508; CARLTON-WIPPERN, K. C., “Analysis of Satellite Collision Probabilities Due to Trajectory and Uncertainties in the Position/Momentum Vectors,” Journal of Space Power, Vol. 12, No. 4, 1993; CHAN, K. F., “Collision Probability Analyses for Earth Orbiting Satellites,” Advances in the Astronautical Sciences, Vol. 96, 1997, pp. 1033-1048; BEREND, N., “Estimation of the Probability of Collision Between Two Catalogued Orbiting Objects,” Advances in Space Research, Vol. 23, No. 1, 1999, pp. 243-247; OLTROGGE, D., and GIST, R., “Collision Vision Situational Awareness for Safe and Reliable Space Operations,” 50th International Astronautical Congress, 4-8 Oct. 1999, Amsterdam, The Netherlands, LAA-99-IAA.6.6.07; AKELLA, M. R., and ALFRIEND, K. T., “Probability of Collision Between Space Objects,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, September-October 2000, pp. 769-772; CHAN, K. F., “Analytical Expressions for Computing Spacecraft Collision Probabilities,” AAS Paper No. 01-119, AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, Calif., 11-15 Feb., 2001; PATERA, R. P., “General Method for Calculating Satellite Collision Probability,” AIAA Journal of Guidance, Control, and Dynamics, Volume 24, Number 4, July-August 2001, pp. 716-722; and ALFANO, S., “Assessing the Instantaneous Risk of Direct Laser Impingement,” Journal of Spacecraft and Rockets, Vol. 40, No. 5, September-October 2003, pp. 678-681).
Likewise, some work has been done to examine accuracy requirements associated with those computations (see GOTTLIEB, R. G., SPONAUGLE, S. J., and GAYLOR, D. E., “Orbit Determination Accuracy Requirements for Collision Avoidance,” AAS Paper No 01-181, AAS/ALAA Space Flight Mechanics Meeting, Feb. 11-15, 2001, Santa Barbara, Calif.; and ALFANO, S., “Relating Position Uncertainty to Maximum Conjunction Probability,” AAS Paper No. 03-548, AAS/AIAA Astrodynamics Specialist Conference, 3-7 Aug., 2003, Big Sky, Mont.).
Typically, one determines if and when a secondary object will transgress a user-defined safety zone. The uncertainties associated with position are represented by three-dimensional Gaussian probability densities. These densities take the form of covariance matrices. For space objects, they can be obtained from the owner-operators or independent surveillance sources such as the US Satellite Catalog (Special Perturbations). When predicting collisions, positions and covariances are propagated to the time of closest approach. Various examples of such prior art collision prediction can be found in U.S. Pat. No. 5,075,694 to Donnangelo et al., U.S. Pat. No. 5,760,737 to Brenner, U.S. Pat. No. 6,694,283 to Alfano et al., U.S. Pat. No. 6,691,034 to Patera et al. and U.S. Pat. No. 6,820,006 to Patera, the details of which are hereby incorporated by reference.
It is possible to find the absolute worst-possible covariance size and orientation that maximizes the probability for a given encounter where the object sizes and rectangular shapes are known. It is also possible to find covariance parameters that maximize the probability for various covariance shapes as determined by the aspect ratio (i.e. the ratio of major-to-minor axes of the projected combined covariance ellipse).
If the maximum probability is below a predefined action threshold, then no further calculations are needed. Such analysis can be insightful even when one only has knowledge of the miss distance and physical object sizes.
Prior art probability computation has been based upon spherical objects. There are many assumptions in this method that reduce the problem's complexity. The physical objects are treated as spheres, thus eliminating the need for attitude information, as illustrated in FIG. 1. For collision analysis, their relative motion is considered linear for the encounter by assuming the effect of relative acceleration is dwarfed by that of the velocity. The positional errors are assumed to be zero-mean, Gaussian, uncorrelated, and constant for the encounter. The relative velocity at the point of closest approach is deemed sufficiently large to ensure a brief encounter time and static covariance. The encounter region is defined when one object is within a standard deviation (σ) combined covariance ellipsoid shell scaled by a factor of n. This user-defined, three-dimensional, n-σ shell is centered on the primary object; n is typically in the range of 3 to 8 to accommodate conjunction possibilities ranging from 97.070911% to 99.999999%.
Because the covariances are expected to be uncorrelated, they are simply summed to form one, large, combined, covariance ellipsoid that is centered at the primary object, as illustrated in FIG. 2. The secondary object passes quickly through this ellipsoid creating a tube-shaped path. A conjunction occurs if the secondary sphere touches the primary sphere, i.e. when the distance between the two projected object centers is less than the sum of their radii. The radius of this collision tube is enlarged to accommodate all possibilities of the secondary touching the primary by combining the radii of both objects.
A plane perpendicular to the relative velocity vector is formed and the combined object and covariance ellipsoid are projected onto this encounter plane. As stated previously, the encounter region is defined by an n-σ shell determined by the user to sufficiently account for conjunction possibilities. Within that shell the tube is straight and rapidly traversed, allowing a decoupling of the dimension associated with the tube path (i.e. relative velocity). The tube becomes a circle on the projected encounter plane. Likewise, the covariance ellipsoid becomes an ellipse, as illustrated in FIG. 3.
The relative velocity vector (decoupled dimension) is associated with the time of closest approach. The conjunction assessment here is concerned with cumulative probability over the time it takes to span the n-σ shell, not an instantaneous probability at a specific time within the shell. Along this dimension, integration of the probability density across the shell produces a number very near unity, meaning the close approach will occur at some time within the shell with near absolute certainty. Thus the cumulative collision probability is reduced to a two-dimensional problem in the encounter plane that is then multiplied by the decoupled dimension's probability. By rounding the latter probability to one, it is eliminated from further calculations.
The resulting two-dimensional probability equation in the encounter plane is given as
                    P        =                              1                          2              ·              π              ·                              σ                x                            ·                              σ                y                                              ·                                    ∫                              -                OBJ                            OBJ                        ⁢                                          ∫                                  -                                                                                    OBJ                        2                                            -                                              y                        2                                                                                                                                                        OBJ                      2                                        -                                          y                      2                                                                                  ⁢                                                exp                  ⁡                                      [                                                                  (                                                                              -                            1                                                    2                                                )                                            ·                                              [                                                                                                            (                                                                                                x                                  +                                                                      x                                    m                                                                                                                                    σ                                  x                                                                                            )                                                        2                                                    +                                                                                    (                                                                                                y                                  +                                                                      y                                    m                                                                                                                                    σ                                  y                                                                                            )                                                        2                                                                          ]                                                              ]                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                                        (        1        )            
where OBJ is the combined object radius, x lies along the major axis, y lies along the minor axis, xm and ym are the respective components of the projected miss distance, and σx and σy are the corresponding standard deviations. For the formulation that follows, the aspect ratio AR is incorporated as a multiple of the minor axis standard deviation (AR≧1) and equation (1) is expressed as
                    P        =                              1                          2              ·              π              ·              AR              ·                                                (                                      σ                    y                                    )                                2                                              ·                                    ∫                              -                OBJ                            OBJ                        ⁢                                          ∫                                  -                                                                                    OBJ                        2                                            -                                              y                        2                                                                                                                                  OBJ                    -                                          y                      2                                                                                  ⁢                                                exp                  ⁡                                      [                                                                  (                                                                              -                            1                                                    2                                                )                                            ·                                              [                                                                                                            (                                                                                                x                                  +                                                                      x                                    m                                                                                                                                    AR                                  ·                                                                      σ                                    y                                                                                                                              )                                                        2                                                    +                                                                                    (                                                                                                y                                  +                                                                      y                                    m                                                                                                                                    σ                                  y                                                                                            )                                                        2                                                                          ]                                                              ]                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                                        (        2        )            
The above equations are also valid for determining the probability of instantaneous line-of-sight impingement. For these instances, the encounter plane is defined perpendicular to the line-of-sight vector and the combined object and covariance ellipsoids projected onto this plane. The axis associated with the line-of-sight vector is then eliminated from the probability calculation. The encounter region is again defined by an n-σ shell determined by the user to sufficiently account for impingement possibilities. As with collision assessment, the instantaneous impingement probability becomes a two-dimensional problem in the encounter plane using equation (2). For the formulation that follows, equation (2) is rewritten as
                    P        =                                            OBJ              2                                      2              ·              π              ·              AR              ·                                                (                                      σ                    y                                    )                                2                                              ·                                    ∫                              -                1                            1                        ⁢                                          ∫                                  -                                                            1                      -                                              y                        2                                                                                                                                  1                    -                                          y                      2                                                                                  ⁢                                                exp                  ⁡                                      [                                                                  (                                                                              -                            1                                                    2                                                )                                            ·                                              [                                                                                                            (                                                                                                                                    x                                    m                                                                    +                                                                      x                                    ·                                    OBJ                                                                                                                                    AR                                  ·                                                                      σ                                    y                                                                                                                              )                                                        2                                                    +                                                                                    (                                                                                                                                    y                                    m                                                                    +                                                                      y                                    ·                                    OBJ                                                                                                                                    σ                                  y                                                                                            )                                                        2                                                                          ]                                                              ]                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                                        (        3        )            